Question

a hospital management system records the wait times for patients checking into the emergency room. based on the records of the management system, the hospital administrator would like to test the claim that the average wait times for patients checking into the emergency room is different than 45 minutes. if the z− test statistic was calculated as z

Answer 1

The hospital administrator can determine whether the average wait times for patients checking into the emergency room are different than 45 minutes by examining the calculated z-test statistic. If the z-test statistic falls outside the critical region corresponding to the chosen significance level (e.g., α = 0.05), the administrator can reject the null hypothesis and conclude that there is a significant difference in the average wait times.

Step-by-step explanation:

To assess the claim that the average wait times for patients checking into the emergency room differ from 45 minutes, a z-test is conducted. The null hypothesis (H₀) posits that the average wait time is 45 minutes, while the alternative hypothesis (H₁) suggests a difference. The z-test statistic is calculated using the formula
`\(z = \frac{\bar{x} - \mu}{(\sigma)/(√(n))}\), where \(\bar{x}\)`is the sample mean, μ is the hypothesized population mean (45 minutes), σ is the population standard deviation, and n is the sample size. The resulting z value is then compared to critical values from the standard normal distribution.

If the calculated z-test statistic falls beyond the critical region (determined by the chosen significance level, e.g., α = 0.05 for a two-tailed test), the null hypothesis is rejected. This implies that there is sufficient evidence to conclude that the average wait times are indeed different from 45 minutes. On the other hand, if the calculated z-value falls within the non-rejection region, there is insufficient evidence to reject the null hypothesis, and the claim of a difference in average wait times is not supported.

In summary, the z-test provides a statistical basis for the hospital administrator to make an informed decision about the claim regarding average wait times in the emergency room, considering both the calculated z-test statistic and the critical values associated with the chosen significance level.

Answer 2

A hospital administrator is testing whether the average wait time in an emergency room differs from the hypothesized 45 minutes, using a z-test. The mean wait time from a sample of 70 patients is 1.5 hours with a standard deviation of 0.5 hours. Z-scores and percentiles are used for understanding the data in relation to the whole set, which is crucial for improving hospital efficiency.

Step-by-step explanation:

Understanding Z-Scores and Wait Times

A hospital management system recorded wait times for patients checking into the emergency room. A hospital administrator is testing the claim that the average wait time is different from 45 minutes. In statistics, a z-test is used to determine if there is a significant difference between the sample mean and the population mean when the population standard deviation is known. In this case, the claim is about the average wait time, which is hypothesized to be 45 minutes, and this is being tested against the sample data.

A specific sample with 70 patients shows an average wait time of 1.5 hours (90 minutes), with a standard deviation of 0.5 hours (30 minutes). If a sample mean substantially differs from the hypothesized population mean (45 minutes), and if this difference is larger than the standard deviations of the wait times, the z-test could indicate a significant difference.

Understanding what percentile a certain wait time falls into, such as being in the 82nd percentile, means that 82% of the wait times were equal to or less than that time. This can provide insight into how one particular wait time compares to the rest of the data set. Using a z-score helps standardize individual data points by comparing them to the mean and standard deviation of the data set, to determine if they are typical or atypical. In the context of a hospital setting, this statistical analysis is vital for improving patient wait times and overall efficiency.

According to the provided data with a confidence interval (CI) of (1.3808, 1.6192), there's an insight into the precision of the average wait time estimate, as this CI represents the range in which the true mean wait time can be expected to fall with a certain level of confidence, often 95%. In the context of a single population mean using the Student's t-distribution, it would be necessary to calculate the t-statistic if the population standard deviation is unknown and the sample size is relatively small.